<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:70_1"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>
 ;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( not $1 is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> $1,<font color="Maroon" title="c1">L</font> );<br/><a NAME="E2:70_1"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c2">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><a NAME="E3:70_1"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 ;<br/><a NAME="E4:70_1"/><i><font color="Green" title="E45">A4</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">Y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:70_1_1"/><i><font color="Green" title="E46">A5</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 <b>and </b><br/><a NAME="E2:70_1_1"/><i><font color="Green" title="E47">A6</font></i>: 
( not <font color="Maroon" title="c4">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">L</font> )
 ;<br/>

<b>assume </b><a NAME="E3:70_1_1"/>
not <font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="yellow_0.html#E1:70_1_1"><i><font color="Green" title="E46">A5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_1_1"/>
( <font color="Maroon" title="c4">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c4">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_1_1_1"/>
<font color="Maroon" title="c4">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:70_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">z</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><b>hence </b><a NAME="E3:70_1_1_1_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T38" target="_self" title="YELLOW_0:th.38">Th38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_1_1_2"/><i><font color="Green" title="E48">A7</font></i>: 
not <font color="Maroon" title="c4">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:70_1_1_1_2"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c6">a</font> = <span class="p1">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c5">z</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D8" target="_self" title="YELLOW_0:def.8">YELLOW_0:def 8</a><br/>

<a NAME="E1:70_1_1_1_2_1"/><i><font color="Green" title="E49">A8</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c5">z</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E1:70_1"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="yellow_0.html#T21" target="_self" title="YELLOW_0:th.21">Th21</a></i>;<br/>
<a NAME="E2:70_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c4">Y</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">a</font> )
 
<b>by </b><i><a class="txt" href="yellow_0.html#E2:70_1_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="yellow_0.html#E1:70_1_1_1_2"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="yellow_0.html#D10" target="_self" title="YELLOW_0:def.10">Def10</a>, <a class="ref" href="yellow_0.html#T19" target="_self" title="YELLOW_0:th.19">Th19</a></i>;<br/>
<a NAME="E3:70_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c4">Y</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c6">a</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c6">a</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T7" target="_self" title="YELLOW_0:th.7">Th7</a>, <a class="ref" href="yellow_0.html#T12" target="_self" title="YELLOW_0:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E4:70_1_1_1_2_1"/><i><font color="Green" title="E50">A9</font></i>: 
<font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T10" target="_self" title="YELLOW_0:th.10">Th10</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>assume </b><a NAME="E1:70_1_1_1_2_1_1"/><i><font color="Green" title="E51">A10</font></i>: 
<font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c7">b</font>
 ;<br/><a NAME="E2:70_1_1_1_2_1_1"/>
( <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:70_1_1_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c4">Y</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:70_1_1_1_2_1_1"><i><font color="Green" title="E51">A10</font></i></a>, <a class="ref" href="yellow_0.html#T9" target="_self" title="YELLOW_0:th.9">Th9</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:70_1_1_1_2_1_1"/><b>then </b>
(  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c5">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="txt" href="yellow_0.html#E2:70_1_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="yellow_0.html#E1:70_1_1_1_2"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="yellow_0.html#E1:70_1_1_1_2_1_1"><i><font color="Green" title="E51">A10</font></i></a>, <a class="ref" href="yellow_0.html#D10" target="_self" title="YELLOW_0:def.10">Def10</a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/><b>hence </b><a NAME="E5:70_1_1_1_2_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:70_1_1_1_2_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="yellow_0.html#T19" target="_self" title="YELLOW_0:th.19">Th19</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c7">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E6:70_1_1_1_2_1"/><i><font color="Green" title="E51">A11</font></i>: 
<font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c7">c</font>
 <b>and </b><br/><a NAME="E7:70_1_1_1_2_1"/><i><font color="Green" title="E52">A12</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">c</font>
 ;<br/>

<a NAME="E8:70_1_1_1_2_1"/><i><font color="Green" title="E53">A13</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E4:70_1_1_1_2_1"><i><font color="Green" title="E50">A9</font></i></a>, <a class="txt" href="yellow_0.html#E7:70_1_1_1_2_1"><i><font color="Green" title="E52">A12</font></i></a></i>;<br/>
<a NAME="E9:70_1_1_1_2_1"/>
( <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:70_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c4">Y</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="yellow_0.html#E6:70_1_1_1_2_1"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="yellow_0.html#T9" target="_self" title="YELLOW_0:th.9">Th9</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:70_1_1_1_2_1"/><b>then </b>
(  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c5">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">c</font> )
 
<b>by </b><i><a class="txt" href="yellow_0.html#E2:70_1_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="yellow_0.html#E1:70_1_1_1_2"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="yellow_0.html#E6:70_1_1_1_2_1"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="yellow_0.html#D10" target="_self" title="YELLOW_0:def.10">Def10</a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E12:70_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E1:70_1_1_1_2_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="yellow_0.html#T19" target="_self" title="YELLOW_0:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E13:70_1_1_1_2_1"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E8:70_1_1_1_2_1"><i><font color="Green" title="E53">A13</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:70_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">X</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="yellow_0.html#E2:70_1"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="yellow_0.html#E3:70_1"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="yellow_0.html#E4:70_1"><i><font color="Green" title="E45">A4</font></i></a>);<br/></i><b>hence </b><a NAME="E6:70_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:70"/>
for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds   <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font>
 ;<br/>

<a NAME="E3:70"/><b>then </b>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">L</font>
 
;<br/>
<b>hence </b><a NAME="E4:70"/>
<font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#T21" target="_self" title="YELLOW_0:th.21">Th21</a></i>;<br/>


</div>
